LEARN · EXPLAIN · REVISE

Explain the projection criterion: a line in a plane is perpendicular to an oblique line exactly when it is perpendicular to its nonzero orthogonal projection.

Read the idea, work independently, then explain what changed.

高一必修第二册(A版).pdf · 8.6 · PDF 153 / printed page 146

TOPIC 01

Perpendicular lines and planes in space

Build understanding of perpendicular lines and planes in space through definitions, contrasting cases and justified applications.

Use definitions, valid spatial reasoning and metric checks for perpendicular lines and planes in space.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

\mathbf v=\mathbf p+\mathbf n

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

Hint 2

Resolve the oblique vector into in-plane and normal components.

Worked solution

Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Calculate or simplify this relation.
$\mathbf u\cdot\mathbf v=\mathbf u\cdot\mathbf p+\mathbf u\cdot\mathbf n=\mathbf u\cdot\mathbf p$
A nonzero projection is required to speak of its direction.

The normal component has zero dot product with every in-plane direction, leaving the projection component.

Checks and common pitfalls: A nonzero projection is required to speak of its direction.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

What must be true before using the main rule for perpendicular lines and planes in space?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use definitions, valid spatial reasoning and metric checks for perpendicular lines and planes in space.
Line-plane test: Perpendicularity to two intersecting in-plane lines proves perpendicularity to the plane.
Angles and distances: Use orthogonal projections for line-plane angles and perpendicular sections for dihedral angles.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Perpendicularity to two intersecting in-plane lines proves perpendicularity to the plane.
Expected reasoning: Use orthogonal projections for line-plane angles and perpendicular sections for dihedral angles.
Expected correction: Perpendicular planes do not make every contained line perpendicular to the other plane.